Question $1-$ Back to the basics

a$)$ Imagine you have a SDOF system as shown in Figure $1-$a$.$ You are asked to imagine an equivalent SDOF with one story similar to Figure $1-$b$.$ Please label the mass and the stiffnesses, i$.$e$.$ fill in the four blanks depicted in Figure $1-$b$!$ Hint: Assume that the columns have the same lateral stiffness.

$($d$)$

$($W$)$

Figure $1$: The SDOF system shown on the left is equivalent to the one on the right.

b$)$ Write down the equation of motion for the SDOF system depicted in Part a$.$

c$)$ Write down the solution for the equation of motion in Part b$,$ i$.$e$.$ find $x(t)!$ Assume you have an initial displacement $x_0$ and initial velocity of $x_0^{\text{'}}$